Simplify. Rewrite the expression in the form $9^n$. $\left(9^2\right)^{5}=$
Answer: $\begin{aligned} \left(9^2\right)^{5}&=9^{2\cdot 5} \\\\ &=9^{10} \end{aligned}$ This follows from the general rule $\left(x^m\right)^{n}=x^{m\cdot n}$. We can also see this is correct by expanding the powers. $\begin{aligned} \left(9^2\right)^{5}&=\underbrace{9^2\cdot 9^2\cdot 9^2\cdot 9^2\cdot 9^2}_\text{5 times} \\\\\\ &=\underbrace{ \underbrace{9\cdot 9}_\text{2 times} \cdot \underbrace{9\cdot 9}_\text{2 times} \cdot \underbrace{9\cdot 9}_\text{2 times} \cdot \underbrace{9\cdot 9}_\text{2 times} \cdot \underbrace{9\cdot 9}_\text{2 times}} _\text{5 times} \\\\ &=9^{10} \end{aligned}$ In conclusion, $\left(9^2\right)^{5}=9^{10}$.